The present invention relates, in general, to data processing with respect to measuring, calibrating, or testing, and, in particular, to filtering or noise reduction/removal.
Any unwanted component in a signal may be viewed as an interferer. To remove an interferer, the signal must first be expanded. Signal expansion methods include Fourier transformation, Taylor series expansion, wavelet transformation, and so on. An expanded signal is then modified to remove the interferer. The inverse of the expansion method is then performed on the expanded and modified signal to obtain the original signal minus the interferer.
If Fourier transformation is used as the expansion method then a time domain signal is converted to the frequency domain, the coefficients of the frequency domain signal are modified, and an Inverse Fourier transform is performed on the modified coefficients to produce, in the time domain, the signal minus the interferer. A Fourier transformation provides spectral accuracy, but may be time consuming or impossible if the signal includes a large amount of data.
A Taylor series expansion is both general and relatively simple to apply, but it is not always easy to determine what effect a modification on a Taylor series expansion coefficient will have on the signal when the inverse of the Taylor series expansion is applied to the modified coefficients.
Another method of removing an interferer from a signal is to multiply each data point in the signal by some factor. However, combinations of pointwise multiplication is less efficient in a domain other than the time domain and it is difficult, if not impossible, to remove a frequency component from a signal by manipulating the signal in the time domain.
Wavelet transformation is a relatively new method of expanding a signal that is being used for data compression and noise reduction. A wavelet transformation involves the use of at least two filters. Typically, the filters are a high-pass filter and a low-pass filter. However, other filters may be used (e.g., Harr, IIR, FIR, bandpass, etc.). The number of samples produced by each filter is half the number of samples of the original signal. Data compression is an inherent benefit of wavelet transformation. The output of each filter may be further wavelet transformed, and Nyquist""s sampling theory sets the upper limit on the number of wavelet transformations that may be nested and still allow the original signal to be completely reconstructed. That is, the number of bits in the output of a filter should be no less than two. In mathematical terms, N=2n, where N is the number of bits, or samples, in the signal to be wavelet transformed, and where n is one plus the maximum number of wavelet transformations that may be nested and still allow the original signal to be completely reconstructed. For example, if a signal consists of 64 bits, or samples, then the maximum number of wavelet transformations that may be nested is five. The number of bits in the outputs of the filters for five nested wavelet transformations are 32, 16, 8, 4, and 2, respectively.
Noise reduction methods that use wavelet transformation typically consist of setting a wavelet transformation coefficient to zero if the coefficient falls below a user-definable threshold. Setting one wavelet transformation coefficient to zero without considering the coefficients that are above the user-definable threshold may introduce local discontinuities in the wavelet transform domain which could cause distortion in the signal upon applying an inverse wavelet transformation to the modified wavelet coefficients.
A signal may be expanded in a wavelet basis to a nesting level of J as follows, where Jxe2x89xa6n, where n is defined as above.       f    ⁢          (      t      )        =                    ∑                  j          =          1                J            ⁢              xe2x80x83            ⁢                        ∑                      k            =            0                                              2                              n                -                j                                      -            1                          ⁢                  xe2x80x83                ⁢                  (                                    d              k              j                        xc3x97                                          Ψ                                  j                  ,                  k                                            ⁢                              (                t                )                                              )                      +                  ∑                  k          =          0                                      2                          n              -              j                                -          1                    ⁢              xe2x80x83            ⁢              (                              s            k            J                    xc3x97                                    ϕ                              j                ,                k                                      ⁢                          (              t              )                                      )            
Wavelet coefficient dkj represents a measure of the change in trend, or transience, in the function f(t) on scale j at position k, and is referred to as a difference. Wavelet coefficient skj represents a measure of the trend in the function f(t) on scale j at position k, and is referred to as an average. These difference and average coefficients are computed recursively as follows, for each j=1,2, . . . ,J and k=0,1, . . . ,2nxe2x88x92jxe2x88x921.             d      k      j        =                  ∑                  l          =          0                          L          -          1                    ⁢              xe2x80x83            ⁢                        g          l                ⁢                  s                      l            +                          2              ⁢              k                        +            1                                j            -            1                                          s      k      j        =                  ∑                  l          =          0                          L          -          1                    ⁢              xe2x80x83            ⁢                        h          l                ⁢                  s                      l            +                          2              ⁢              k                        +            1                                j            -            1                                          s      k      0        =          f      ⁢              (                  t          k                )            
U.S. Pat. No. 4,853,903, entitled xe2x80x9cMETHOD AND APPARATUS FOR REMOVING SINUSOIDAL NOISE FROM SEISMIC DATA,xe2x80x9d discloses a device for and method of removing an interferer from a signal by using Fourier transformation. The device and method of U.S. Pat. No. 4,853,903 is computationally intense for, and may not work on, signals that include a large number of samples, whereas the present invention is less computationally intense for, and would work on, such signals. U.S. Pat. No. 4,853,903 is hereby incorporated by reference into the specification of the present invention.
U.S. Pat. No. 5,974,181, entitled xe2x80x9cDATA COMPRESSION SYSTEM, METHOD, AND APPARATUS,xe2x80x9d discloses a device for and method of compressing a signal using wavelet transformation and, in an alternate embodiment, Fourier transformation. However, U.S. Pat. No. 5,974,181 does not disclose the method of removing an interferer from a signal as does the present invention. U.S. Pat. No. 5,974,181 is hereby incorporated by reference into the specification of the present invention.
U.S. Pat. No. 6,014,412, entitled xe2x80x9cDIGITAL RADIO FREQUENCY INTERFERENCE CANCELLER,xe2x80x9d discloses a device for and method of removing an interferer from a signal by working with the signal in the frequency domain, obtaining an estimate of the interferer, and subtracting the estimate from the signal. The method of U.S. Pat. No. 6,014,412 is computationally intense for, and may not work on, signals that include a large number of samples, whereas the present invention is less computationally intense for, and would work on, such signals. U.S. Pat. No. 6,014,412 is hereby incorporated by reference into the specification of the present invention.
It is an object of the present invention to remove a feature from a signal.
It is another object of the present invention to remove a feature from a signal by a combination of wavelet and Fourier transformation.
It is another object of the present invention to remove a frequency from a time domain signal by reducing the samples in the signal by wavelet transformation, projecting the feature to be removed onto a wavelet basis using Fourier transformation and subtracting the projection of the feature from the projection of the original time domain signal.
The present invention is a method of removing at least one user-definable frequency from a signal in a manner that is, computationally, more efficient than prior art methods and may be applied to signals containing a number of samples that cannot be processed by prior art methods.
The first step of the method is receiving a signal in the time domain.
The second step of the method is identifying at least one frequency to be removed from the signal.
The third step of the method is identifying a possible number of resolution levels to which the signal will be reduced using wavelet transformation.
The fourth step of the method is computing at least one wavelet difference coefficient for each of the number of resolution levels identified in the third step and at least one wavelet average coefficient for the number of resolution levels that is at the lowest resolution level (i.e., numerically last).
The fifth step of the method is creating a projection of each at least one frequency selected in the second step onto the lowest, or numerically last, of the resolution levels identified in the third step using Fourier transformation.
The sixth step of the method is subtracting the result of the fifth step from the corresponding at least one wavelet average coefficient computed in the fourth step.
The seventh, and last, step of the method is adding the results of the sixth step to the at least one wavelet difference coefficient computed in the fourth step. The result of the seventh step is the received signal minus the at least one user-definable frequency.